ACT (Enhanced)MathematicsHard
A scientist is tracking the population of a certain species of fish in a lake. The population, P(t), after t years, follows a logistic growth model given by P(t) = 1000 / (1 + 9e^(-0.2t)). What is the carrying capacity of the lake for this fish species?
- A1000 fish
- B9 fish
- C100 fish
- D9000 fish
Show answer & explanationAnswer & explanation
Correct answer: A. 1000 fish
In a logistic growth model of the form P(t) = K / (1 + Ae^(-kt)), K represents the carrying capacity. In the given function P(t) = 1000 / (1 + 9e^(-0.2t)), the value of K is 1000. This is the maximum population the environment can sustain.
Why the other options are wrong
- B. This is related to the 'A' constant in the denominator, not the carrying capacity.
- C. Incorrect identification of the carrying capacity.
- D. Incorrect identification; this would be (A+1)*K or similar error.
Logistic Growth Carrying Capacity
In a logistic growth model, the carrying capacity (K) is the maximum population size that the environment can sustain indefinitely, represented by the numerator of the logistic function.
- The population approaches K as time approaches infinity.
- It's the upper horizontal asymptote of the logistic curve.
- Limits resources and environmental factors determine K.
Memory trick: The 'K'ap is 'K'ing at the 'K'arriage 'K'apacity.